Math 101—Midterm Exam #2, Practice Midterm B
Duration: 50 minutes
Surname (Last Name)
Given Name
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Do not open this test until instructed to do so! This exam should have 8 pages, including this cover sheet.
No textbooks, notes, calculators, or other aids are allowed; phones, pencil cases, and other extraneous items
cannot be on your desk. Turn off cell phones and anything that could make noise during the exam.
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be given
for the answer without the correct accompanying work. Problems 5–7 are long-answer: give complete
arguments and explanations for all your calculations—answers without justifications will not be marked.
Continue on the back of the page if you run out of space.
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Problem Out of Score Problem Out of Score
1
6
5
8
2
6
6
8
3
6
7
8
4
3
Total
45
Math 101—Midterm Exam #2, Practice Midterm B
page 2 of 8
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
1a. [3 pts] Determine whether the following sequences converge or diverge. If they converge,
find the limit
n
(a) an = √sin
n+1
(b) bn = ln(n + 1) − ln n
(c) cn = cos( πn
)
2
1b. [3 pts] Determine whether the improper integral
Z ∞
4 + sin x
p
dx
x − 1/2
1
converges or not. If it converges, find its value.
Math 101—Midterm Exam #2, Practice Midterm B
page 3 of 8
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
2a. [3 pts] Use Simpson’s Rule with n = 4 to approximate the value of the integral
Z 5
(2x2 + 3x) dx.
−3
Z
2b. [3 pts] Evaluate
0
π/4
tan3 x sec2 x dx. Simplify your answer fully.
Math 101—Midterm Exam #2, Practice Midterm B
page 4 of 8
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
3a. [3 pts] Find the general antiderivative of f (x) = x2 sin(πx).
Z
3b. [3 pts] Evaluate
1
2
√
x2
dx.
x2 − 1
Math 101—Midterm Exam #2, Practice Midterm B
page 5 of 8
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
Z 4
x−1
4. [3 pts] Evaluate
dx.
2
0 x − 4x − 5
Math 101—Midterm Exam #2, Practice Midterm B
page 6 of 8
Problems 5–7 are long-answer: give complete arguments and explanations for all your calculations—
answers without justifications will not be marked.
5.
(a) [4 pts] A lamina with density ρ = 3 is in the shape of the region under the graph of
y = e2x between x = −1 and x = 1. Find My , the moment with respect to the y-axis, of
this lamina.
(b) [4 pts] Evaluate the integral
Z
√
(x + 1) x2 + 2x + 2 dx.
Math 101—Midterm Exam #2, Practice Midterm B
page 7 of 8
6.
(a) [4 pts] Use integration to calculate the area of the ellipse
x2 y 2
+
=1
4
9
Hint: exploit the symmetry of the ellipse with respect to both axes.
6b. [4 pts] Determine whether the integral
Z e3
ln x dx
0
converges or diverges. If it converges, find its value.
Math 101—Midterm Exam #2, Practice Midterm B
7.
(a) [8 pts] Find the solution to the differential equation
dy
y
= 4
dx
x + x2
which satisfies the initial condition y(1) = 1/e.
page 8 of 8